Transformations of Functions - Infinity is Really Big

Calculus 1, Lectures 6A through 7B

*Visual for the fact that $\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1$ in terms of transformations of functions.*

*Visual for the fact that $\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1$ in terms of transformations of functions.*

In the 1600’s, algebra and geometry got “married” to create a new subject called analytic geometry. Insights in either branch led to insights in the other. This has been very fruitful over the years.

The symbiosis between these subjects can be especially helpful when we consider transformations of functions in certain situations. Two of the most interesting situations to consider are exponential functions and logarithmic functions.

As usual in this series of blog posts about my fall 2019 Calculus 1 lectures at Bethel University, let’s start with a summary of some lectures. In this case, we summarize the content of lectures 6A through 7B.

Lecture 6A

In Lecture 6A I start with a brief discussion about actuarial careers. Actuaries need to know about mathematical finance. Thus, this provides a jumping point for the next slide in the lecture, which is a problem about using the quadratic formula in finance.

The situation is that there is a two-coupon bond for which we would like to know the yield rate. The yield rate is a measure of how much return we are getting on our investment in the bond. The procedure used is based on the compound interest formula, as well as the idea of equating future values of cash in and cash out.

Calculus 1, Lec 6A: Actuaries, Yield Rate of a Coupon Bond, Transform y=x^2, Logarithm Properties

From there, I go on to discuss transformations of functions and their graphs, but in the context of the quadratic function $y=f(x)=x^{2}$ rather than logarithmic functions. This is includes important visuals made in the CAS program Mathematica, by Wolfram Research.

Finally, I briefly review properties of logarithms in anticipation of proving one of them in Lecture 6B.

Lecture 6B

I begin Lecture 6B with a proof of the following fact. If $b>0$ , $b\not=1$ , and $A,B>0$ , then $\log_{b}(AB)=\log_{b}(A)+\log_{b}(B)$ .

This is sometimes called the “product-to-sum” property of logarithms. Its proof requires us to use a corresponding property of exponents. That property is that $b^{x+y}=b^{x}\cdot b^{y}$ . You could call this a “sum-to-product” property of exponential functions.

Calc 1, Lec 6B: Logarithm Proof, Graphs of Logs, Cosine & Sine Unit Circle Def, Sinusoidal Functions

It is at time 9:14 of this video that I talk about the main thing that I want to emphasize in this blog post further below. At that point of the lecture, I discuss and visualize the equivalence between horizontal compressions and vertical translations for the graphs of logarithms. It is based on the product-to-sum property written above.

Finally, the unit circle definitions of cosine and sine, as well as the definition of a general sinusoidal function are given in Lecture 6B. The important concepts of period, frequency, amplitude, and phase angle for a sinusoidal function are discussed as well.

Lecture 7A

At the beginning of Lecture 7A I do another proof. This is a true “mathematical analysis-type” proof involving a limit concept. The goal is to prove the following facts.

$\displaystyle\lim_{x\rightarrow +\infty}\log(x)=+\infty\mbox{ and } \displaystyle\lim_{x\rightarrow 0^{+}}\log(x)=-\infty$

Intuitively, the idea of the proof of the first fact is to show that $\log(x)$ eventually gets above (and stays above) any horizontal line $y=M$ when $x$ is big enough. For the second fact, we must show that $\log(x)$ eventually gets below (and stays below) any horizontal line $y=-M$ when $x$ is close enough to zero.

Calculus 1, Lec 7A: Proof That log(x) Goes to Infinity, Log Plots, Graphs of Cosine, Sine, & Tangent

After this proof, which is surprisingly short, I move on from pure mathematics to an important topic from applied mathematics. That topic is curve-fitting (of data). In this case, it’s based on data about the United States Gross Domestic Product (GDP) from 2009 through 2018. I also discussed this in Lecture 4A, which you can find embedded in my blog post titled “The Derivative of x^2”.

The main point of the curve fitting is to use the idea of a log plot (or semi-log plot) to confirm that the data are well-modeled by an exponential function. I also discuss the theoretical justification for this method, which relies on the use of properties of logarithms, such as the product-to-sum property.

Finally, the trigonometric topics from Lecture 6B are reviewed in anticipation of going into more depth in Lecture 7B.

Lecture 7B

The major part of Lecture 7B is a very in-depth problem about modeling sinusoidal behavior and solving for all solutions of a sinusoidal equation using the inverse sine function. I also describe how to use the unit circle to help find all the solutions, though I make a number of verbal mistakes as I go. These verbal mistakes do not translate to written mistakes however. The final answers are correct.

Calc 1, Lec 7B: Sinusoidal Problem Solving (Inverse Trig Functions), Polynomial Functions

You should realize as you watch this that the ranges of inverse trigonometric functions are chosen by convention (agreement among mathematicians). This convention affects how technology, such as calculators, are programmed. However, different conventions could have been used.

The range of $y=\cos^{-1}(x)=\arccos(x)$ is $[0,\pi]=\{y\ |\ 0\leq y\leq \pi\}$ . The range of $y=\sin^{-1}(x)=\arcsin(x)$ is $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]=\{y\ |\ -\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\}$ . And the range of $y=\tan^{-1}(x)=\arctan(x)$ is $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]=\{y\ |\ -\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\}$ .

Finally, Lecture 7B includes a very brief discussion of polynomial functions and their properties. It was the best I could do in 10 minutes time. In Lecture 8A, I am more systematic about the subject.

Back to Transformations of Functions

Now I go back and take the time to explore the content emphasized in Lectures 6A and 6B in more depth. Specifically, you should watch Lecture 6B starting at time 9:14 as mentioned above.

Let me start by picking something specific for $b$ , such as $b=2$ . I will also choose to use slightly different notation from above and instead write the product-to-sum property as $\log_{2}(ax)=\log_{2}(a)+\log_{2}(x)$ , where $a,x>0$ .

If we let $y=f(x)=\log_{2}(x)$ , this property implies that $f(ax)=f(x)+f(a)$ . Suppose we think of $a>0$ as a fixed number and $x>0$ as a variable.

If $a>1$ , then the graph of $y=f(ax)$ represents a horizontal compression, by a factor of $a$ , of the graph of $y=f(x)$ toward the vertical $y$ -axis. This means that all the points on the graph of $y=f(x)$ move $a$ -times closer to the $y$ -axis to obtain the graph of $y=f(ax)$ .

On the other hand, if $a>1$ , then $f(a)=\log_{2}(a)>0$ and the graph of $y=f(x)+f(a)$ represents an upward vertical translation (shift), by the amount $f(a)$ , of the graph of $y=f(x)$ away from the horizontal $x$ -axis.

The fact that $f(ax)=f(x)+f(a)$ means these two transformations of the graph of the original function $y=f(x)$ produce the exact same graph.

This is a pretty amazing thing, but you should realize that it only works for logarithmic functions.

Animation Demonstrating This Equivalence

The animation below illustrates this in the case where $a=2$ . Note that $\log_{2}(2)=1$ , so the equation being discussed becomes $f(2x)=f(x)+f(2)=f(x)+1$ . In other words, a horizontal compression by a factor of 2 produces the same graph as a vertical upward translation by 1 unit.

Illustrating transformations of functions for a logarithm. This shows that a horizontal compression is equivalent to a vertical translation. — Visual for the fact that $\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1$ . The blue vertical lines all have the same length (of 1 when $a$ reaches 2). The red horizontal lines do not all have the same length. They represent the fact that the points on the black graph are moving to points that are twice as close to the vertical $y$ -axis when $a$ reaches 2.

Related Property for Exponential Functions

The logarithmic product-to-sum identity $\log_{b}(AB)=\log_{b}(A)+\log_{b}(B)$ is equivalent to the exponential sum-to-product identity $b^{c+d}=b^{c}\cdot b^{d}$ . This last property can also be thought of in terms of transformations of functions.

If $f(x)=b^{x}$ , then we can write the sum-to-product identity as $f(x+a)=f(a)f(x)$. In other words, an appropriate horizontal translation by $|a|$ units (to the right if $a<0$ and to the left if $a>0$ ) is equivalent to a vertical stretch/compression away from the horizontal $x$ -axis by a factor of $f(a)=b^{a}$ . This is a vertical stretch if $a>0$ and a vertical compression if $a<0$ .

If $b=2$ and $a=1$ , then the property becomes $f(x+1)=f(1)f(x)=2f(x)$ . In other words, a horizontal translation to the left by 1 units is equivalent to a vertical stretch by a factor of 2. This is illustrated below.

Illustrating transformations of functions for an exponential function. This shows that a vertical stretch is equivalent to a horizontal translation. — Visual for the fact that $2^{x+1}=2\cdot 2^{x}$ . The red horizontal lines all have the same length (of 1 when $a$ reaches 1). The blue vertical lines do not all have the same length. They represent the fact that the points on the black graph are moving to points that are twice as far from the horizontal $x$ -axis when $a$ reaches 1.

Visual for the fact that $2^{x+1}=2\cdot 2^{x}$ . The red horizontal lines all have the same length (of 1 when $a$ reaches 1). The blue vertical lines do not all have the same length. They represent the fact that the points on the black graph are moving to points that are twice as far from the horizontal $x$ -axis when $a$ reaches 1.

It is also possible to interpret many trigonometric identities in terms of transformations of functions. A simple example is $\sin\left(x+\frac{\pi}{2}\right)=\cos(x)$ . A more complicated example is $\sin(x+a)=\cos(a)\sin(x)+\sin(a)\cos(x)$ .

I encourage you to explore this on your own.